Identifying a polytope by its fibre polytopes (Q2022358)

For a convex polytope \(P\) in a \(d\)-dimensional Euclidean vector space \(V\) and for \(L\in{\mathcal G}_r(V)\), the Grassmannian of \(r\)-dimensional linear subspaces of \(V\), the author defines the fibre polytope of \(P\) in \(L\) by \[\mathrm{fib}(P;L)=\int_{\Psi P} \left((P-z)\cap L\right)dz,\] where \(\psi\) is the orthogonal projection from \(V\) to \(L^\perp\). The main result says that if \(2\le r < d\), then a full-dimensional polytope \(P\) is uniquely determined by its fibre polytopes \(\mathrm{fib}(P;L)\), \(L\in {\mathcal G}_r(V)\). The proof first finds the \((r-1)\)-skeleton of the normal fan of \(P\) and then uses an inductive procedure, together with Minkowski's existence theorem, to find the faces of \(P\). It is also shown that if \(r\ge 4\) and the \(r\)-dimensional fibre polytopes of \(P\) are zonotopes, then \(P\) is a zonotope.